SENSORLESS FIELD ORIENTED CONTROL OF BRUSHLESS ...

x kCx
(3.75)
abc
The set of variables defined by x abc are the phase variables. When transformed by the matrix C
they are written as x , which is a vector defined in the stationary reference frame. Given
Equation (3.72), x (which exists in the complex/αβ plane) could be said to be in the stationary
reference frame as well. The matrix C goes by many names. In the popular literature and in this
report, C is called the Clarke transform (after Edith Clarke), even though this is not the original
transform (discussed in Appendix D).
From prior discussion is should be clear how the definition of Equation (3.69) came about: a
vector is composed by summing the product of a scalar with a basis vector and in SV theory there
are three complex basis vectors, one aligned to each phase axis. By simply substituting the
rectangular form of the basis vectors into Equation (3.69), it is clear that Equation (3.70) is
identical. Equation (3.70) makes sense from a cursory examination of projection as well. Using
simple trigonometry, all of a projects onto α, 1/2 of b and c project onto α, 3/2 of b
projects onto β and the same of c projects onto negative β. Then when Equation (3.72) is
considered, the real-valued matrix equivalent (Equation 3.74) is evident. 24 This shows that the SV
definition is a transform (given by Equation 3.74) and that the SV is a complex-valued
representation of a vector (Equations 3.72 and 3.73). This result is important because it is one
point of connection (among several) between the traditional matrix-based analysis of polyphase
circuits and machines ([11], [12], [32], [34]) (or tensor analysis [13]) and more modern “complex
analysis” [78], [16], [15], [87].
Defining the phase basis vectors in αβ coordinates (Equation 3.76) allows Equation (3.74) to be
written as Equation (3.77).
1 1/2 1/2
a ; ;
0
b c (3.76)
3/2 3/2 24 Historically the development proceeded in the opposite direction. A transform similar to the original
Clarke transform was used as early as 1917 [11, p.310] and then around 1954 complex space vector theory
was introduced [16], [15]. Finally, somewhere in that history was the interaction between matrix
descriptions of machines and circuits and the application of tensor analysis (as early as 1938). The details
are unimportant here. The best intuition can be had when the SV is viewed as a vector but the matrix case is
easier to use to obtain key results.
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